The Jacobian module, the Milnor fiber, and the D-module generated by fs

• 成果类型: Article
• 署名作者: Walther, Uli
• 署名单位: Purdue University System; Purdue University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-016-0684-2
• 发表日期: 2017
• 页码: 1239-1287
• 关键词: logarithmic comparison theorem; hyperplane arrangements; free divisors; zeta-functions; differential-operators; b-functions; complexes; IDEALS; singularity; Duality

摘要

For a germ f on a complex manifold X, we introduce a complex derived from the Liouville form acting on logarithmic differential forms, and give an exactness criterion. We use this Liouville complex to connect properties of the D-module generated by to homological data of the Jacobian ideal; specifically we show that for a large class of germs the annihilator of is generated by derivations. Through local cohomology, we connect the cohomology of the Milnor fiber to the Jacobian module via logarithmic differentials. In particular, we consider (not necessarily reduced) hyperplane arrangements: we prove a conjecture of Terao on the annihilator of 1 / f; we confirm in many cases a corresponding conjecture on the annihilator of but we disprove it in general; we show that the Bernstein-Sato polynomial of an arrangement is not determined by its intersection lattice; we prove that arrangements for which the annihilator of is generated by derivations fulfill the Strong Monodromy Conjecture, and that this includes as very special cases all arrangements of Coxeter and of crystallographic type, and all multi-arrangements in dimension 3.